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A straightforward scheme to guarantee the fair-coexistence between Wi- Fi and cellular is to let cellular adopt a discontinuous, duty-cycle transmission pattern, which is also know as LTE-U [73, 87]. Specifically, within a fixed transmission interval, cellular network transmits for a fraction η of time (0 ≤ η ≤1), and is muted for the complementary 1-η fraction.

The cellular transmission duy cycle η can be fixed or adaptively ad- justed based on Wi-Fi medium utilization [73]. Generally, η needs to be chosen in such a way that cellular shall not impact Wi-Fi more than an additional Wi-Fi network w.r.t. SINR coverage probability, rate coverage, etc. We con- sider a static muting pattern for cellular, where all the BSs follow the same muting pattern either synchronously or asynchronously. If the BSs are muted synchronously, they transmit and mute at the same time. If the BSs are muted asynchronously, the neighboring BSs could adopt a shifted version of the muting pattern [66]. For simplicity, we assume each BS is transmitting with probability η at a given time under the asynchronous scheme. In the rest of this section, the time-averaged DST and rate coverage performance when cellular transmits discontinuously are derived.

3.5.1 Cellular Network with Synchronous Discontinuous Transmis- sion Pattern

In this case, since all BSs transmit and mute at the same time, the MAP for the tagged Wi-Fi AP during cellular “On” and “Off” period are ˆ

pW

0,M AP(λW, λL) and ˆp0,M APW (λW,0) respectively, where ˆpW0,MAPis given in (3.7).

Similarly, the SINR coverage probability of the typical Wi-Fi STA (resp. cellular UE) with threshold T is pW

0 (T, λW, λL) (resp. pL0(T, λW, λL)) and

pW

0 (T, λW,0) (resp. 0) during cellular “On” and “Off” period respectively,

where pW

0 and pL0 are provided in Lemma 3.4.4 and Lemma 3.4.5. Define the

time-averaged DST with SINR threshold T as the time-averaged fraction of links that can support SINR level T .

Lemma 3.5.1. When cellular network adopts a synchronous discontinuous transmission pattern with duty cycle η, the time-averaged DST with threshold T for the Wi-Fi and cellular network are given by:

dW_1,suc(λW, λL, T, η) = ηλWpˆ0,M APW (λW, λL)pW0 (T, λW, λL)

+ (1 − η)λWpˆW0,M AP(λW,0)pW0 (T, λW,0),

dL_1,suc(λW, λL, T, η) = ηλLpL0(T, λW, λL), (3.11)

respectively.

Since cellular transmits for η fraction of time and silences for 1 − η fraction time, Lemma 3.5.1 can be obtained directly from Definition 3.3.1.

The time-averaged rate coverage probability with threshold ρ is defined as the time-averaged fraction of BSs/APs that can support an aggregate data

rate of ρ. Since each cellular BS transmits for η fraction of time, we treat the MAP of the tagged BS as η in (3.4).

Lemma 3.5.2. When cellular network adopts a synchronous discontinuous transmission pattern with duty cycle η, the time-averaged rate coverage prob- ability with rate threshold ρ for Wi-Fi and cellular are given by:

P_1,rateW (λW, λL, ρ, η) =ηpW0 (2 ρ B ˆpW_{0,M AP}(_{λW ,λL)} − 1, λW, λL) + (1 − η)pW0 (2 ρ B ˆpW_{0,M AP}(_{λW ,0)} − 1, λW,0), P_1,rateL (λW, λL, ρ, η) =pL0(2 ρ Bη − 1, λ W, λL), (3.12) respectively.

Proof. Please see Appendix 3.9.6.

It is straightforward from (3.11) and (3.12) that better DST and rate coverage can be achieved by Wi-Fi when η decreases. By contrast, since pL₀(T, λW, λL) is a decreasing function w.r.t. the SINR threshold T , cellular

achieves better DST and rate coverage when η increases.

3.5.2 Cellular Network with Asynchronous Discontinuous Trans- mission Pattern

Since each BS transmits independently with probability η at a given time, the BSs contributing to the interference of Wi-Fi form a PPP with intensity ηλL. Therefore, the MAP for the tagged AP is ˆpW0,MAP(λW, ηλL), and

is pW

0 (T, λW, ηλL). Correspondingly, the time-averaged DST of Wi-Fi is given

by:

dW_2,suc(λW, λL, T, η) = λWpˆW0,M AP(λW, ηλL)pW0 (T, λW, ηλL), (3.13)

and the time-averaged rate coverage probability of Wi-Fi is given by:

P_2,rateW (λW, λL, ρ, η) = pW0 (2

ρ B ˆpW_{0,M AP}(_{λW ,ηλL)}

− 1, λW, ηλL). (3.14)

According to (3.13) and (3.14), Wi-Fi achieves better DST and rate coverage when η decreases.

For cellular network, during the η fraction of time that the tagged BS transmits, the interfering BSs form a PPP with intensity ηλL. Thus, the

time-averaged DST of cellular is given by:

dL_2,suc(λW, λL, T, η) = λLη

Z ∞

0

pL₀(r0, T, λW, ηλL)2πλLr0exp(−λLπr02)dr0,

(3.15) and the time-averaged rate coverage probability is given by:

P_2,rateL (λW, λL, ρ, η) = Z ∞ 0 pL₀(r0,2 ρ Bη − 1, λ W, ηλL)2πλLr0exp(−λLπr20)dr0, (3.16) where pL 0(r0, T, λW, λL) is derived in Lemma 3.4.5.

3.5.3 Comparison of Synchronous and Asynchronous Muting Pat- terns

Fig. 3.5 and Fig. 3.6 show the analytical time-averaged DST and rate coverage performance when λW = 400 APs/km2 and λL = 400 BSs/km2. In

SINR Threshold (dB)

-10 -5 0 5 10 15 20

Average number of

successful links per km

2 0 50 100 150 200 250 300 η = 33.3%, Synchronous η = 33.3%, Asynchronous η = 50%, Synchronous η = 50%, Asynchronous η = 66.7%, Synchronous η = 66.7%, Asynchronous (λ W, λL) = (400,400)

(a) DST for Wi-Fi network

SINR Threshold (dB)

-10 -5 0 5 10 15 20

Average number of

successful links per km

2 0 50 100 150 200 250 300 η = 66.7%, Asynchronous η = 66.7%, Synchronous η = 50%, Asynchronous η = 50%, Synchronous η = 33.3%, Asynchronous η = 33.3%, Synchronous (λ_W, λ_L) = (400,400)

(b) DST for cellular network

Figure 3.5: DST comparison.

terms of Wi-Fi DST and rate coverage performance, the synchronous cellular muting pattern generally outperforms the asynchronous one. This is due to fact that when all cellular BSs are muted, Wi-Fi APs observe a much cleaner channel and therefore benefit more compared to the asynchronous scheme. Since cellular interferers form an independent thinning of the BS process un- der the asynchronous muting pattern, the latter outperforms the synchronous pattern in terms of DST and rate coverage. In addition, Fig. 3.5 and Fig. 3.6 also indicate that cellular needs to adopt a short transmission duty cycle η (e.g., less than 50%) to protect Wi-Fi. However, cellular is also more sensitive to the transmission duty cycle compared to Wi-Fi, which means that a very small η leads to much degraded performance of cellular. Therefore, a syn- chronous muting pattern with a reasonably short cellular transmission duty cycle (e.g., within 33.3% to 50%) is suggested to best protect Wi-Fi.

Rate Threshold (Mbps)

0 10 20 30 40 50 60 70 80

Rate Coverage Probability

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 η = 33.3%, Synchronous η = 33.3%, Asynchronous η = 50%, Synchronous η = 50%, Asynchronous η = 66.7%, Synchronous η = 66.7%, Asynchronous (λW, λL) = (400,400)

(a) Rate coverage for Wi-Fi network

Rate Threshold (Mbps)

0 10 20 30 40 50 60 70 80

Rate Coverage Probability

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 η = 66.7%, Asynchronous η = 66.7%, Synchronous η = 50%, Asynchronous η = 50%, Synchronous η = 33.3%, Asynchronous η = 33.3%, Synchronous (λW,λL) = (400, 400)

(b) Rate coverage for cellular network

Figure 3.6: Rate coverage comparison.